The Method of Harmonic Mapping of Regions with a Notch

It is well known that common numerical methods encounter serious computational difficulties when constructing harmonic mappings of regions with notches and generating computational grids in such regions on the basis of these mappings. We propose an efficient computational technique for constructing a harmonic mapping of domains with a rectangular notch based on the analytical–numerical multipole method. Our research demonstrates high computational efficiency of the proposed method; the use of only $40$ approximative functions (multipoles) provides an error of less than $10^{-7}$ in the $C (\overline{g})$ norm. The result is obtained with the help of a posteriori estimation. We find a condition ensuring that the grid line issuing from the vertex of an angle of magnitude $\pi \beta$, $\beta >1$, is not tangent to the angle sides at this vertex, which prevents the emergence of an adverse grid self-overlapping effect.